If you have access to phpMyAdmin, you should have all the tools necessary to specify foreign key constraints (migrations aren't required for that). Not doing so is ill-advised, since foreign keys ensure the consistency of your data. Using them, you can guarantee that all comments refer to a legitimate post at all times.

Make sure both tables use the InnoDB engine, add a post_id column to the comments table, define a foreign key relationship, and consider how you want updates and deletes to be handled (cascade, restrict, etc).

I didn't answer your original question of "Will Eloquent handle relationships without foreign keys defined?" because you shouldn't proceed down that path. I would suggest learning how to define foreign keys in phpMyAdmin. If there's a lot of existing data with inconsistencies that prevent you from defining foreign keys, spend time cleaning it up first.